 hello friends so let's take up a question on bearings in this question it's given that a ship sails 22 kilometers from a point a on a bearing of zero three zero and a further 30 kilometer on a bearing of zero nine zero to arrive at b so you can see the positions of a here is a here is b now ac is 22 kilometers bearing is 30 degrees and from here you can see bearing 90 degrees so what is bearing basically you have to take the north direction and from there you have to measure the angle so first of all this north and 30 degrees so hence ac is the first diagram and for at point c again 90 degree because in this case at point c the bearing is zero nine zero so hence it will move towards east and b is due east to c so i hope how to make the diagram is clearer to you so you have to find out the length a b this one a b a b and you have to also find out the bearing of a b that means you have to find out angle let's say n a b so this angle is theta is unknown this is what you have to find out right b from a that's the bearing of b from a so let's start so if you see let me just drop up a pendicular here okay so this is the perpendicular so let me call it f this point as f now let's solve so how do we go about it so if you see a b how to find out a b if we know ad and bd bd is a perpendicular dropped here you can see bd is perpendicular so by pythagoras theorem by pythagoras theorem pythagoras theorem we can say a b square is equal to ad square plus bd square right what will be ad and bd let's see so if you see ad can be written as af plus fd whole square plus bd square right bd square this is the thing now let us consider triangle triangle acf where angle f is 90 degrees isn't it now we can use trigonometry to solve this so acf is 90 so hence can i not write cf cf by ac is cf by ac right so if that is 30 degree guys can i not say this angle here is 60 degrees right if the bearing is 30 so n angle n ac is 30 degrees so angle f ac will be 90 degrees minus 30 degrees which is 60 degrees clear right so hence cf by ac will be nothing but sine 60 degrees opposite by hypotenuse so sine 60 is nothing but root 3 by 2 from our trigonometric table we know this so what is cf guys cf is equal to ac into root 3 by 2 now thankfully ac is known to us which is 22 kilometers so 22 into root 3 by 2 which is 11 root 3 kilometers no doubts about it so we got cf now if you look closely cf is equal to bd actually so this is equal to bd because cbdf is a parallelogram cbdf sorry cbdf is a parallelogram right there are 490 degrees right so hence it's a parallelogram now so that's that being clear now we have to find out af what will be af guys so if you see clearly again you can use af upon ac af upon ac is cos 60 degrees correct af upon ac is cos 60 which is half right cos 60 half this means af is known af is equal to ac into half right now ac was 11 so it will be 22 into half that is 11 kilometers fair enough so we got af as well as fc now we can come with or we can solve this question so we now know ab square is equal to ad square plus bd square we just saw above which is now equal to ad can be written as af plus fd whole squared plus bd squared isn't it now af can be written as af we already know now af is or af is 11 kilometers isn't it so af is 11 and fd fd guys is equal to bd or sorry bc which is 30 kilometers isn't it see this length fd is equal to this length clearly 30 kilometers right so 11 plus 30 because this is a parallelogram or a rectangle whichever way so 11 plus 30 squared plus bd squared bd was root 3 by sorry 11 root 3 so 11 root 3 whole squared so let's so this is 40 30 plus 1 41 so let's find out this 41 square is 1681 1681 41 square is 1681 plus 11 square into 3 so and 121 into 3 is 363 okay 121 and so hence now add both of them 1681 plus 363 is 2044 right so now ab is clearly under root 2044 is it so once again check the calculations if check calculations are okay so hence 2044 is the total is the square of bd so hence now we can find out square root of 2044 comes out to be 45.21 kilometers okay so this is how much the point b is from a now we have to find out so ab is now known we have to find out bearing so how to find out bearing so angle b ab right angle b ad we have to find out is nothing but or rather if you see angle b ad tan of angle b ad is nothing but if you see from the diagram it is bd upon ad right what was bd guys bd was 11 root 3 and what was ad guys ad was 11 plus 30 41 so it is 11 root 3 by 41 right so angle b ad is equal to tan inverse 11 root 3 by 41 this is b ad so but you have to find out the bearing so bearing is n ab isn't it this angle will be the bearing theta right so theta or the bearing bearing bearing of b with respect to a will be how much 90 degrees minus angle b ad so which is nothing but 90 degrees minus tan inverse 11 root 3 by 41 right so i'm not finding the values over here so it's not required you know the concept now you can use trigonometric tables to find out the values and the this is the answer so now you know how to find out the bearings of any particular point with respect to other point right what is the what is the concept of bearing bearing is nothing but the angle made by the line joining the point with respect to you want to find out the bearing and the north direction right so that's what is the bearing so clockwise direction if you measure the angle from north of any particular point from a given point then we know that is the bearing okay so i hope you understood this problem